op.update(kwargs)
# --- Initialise mesh
swp = AdaptiveProblem(op)
# Refine around equator and/or soliton
if initial_monitor is not None:
mesh_mover = MeshMover(swp.meshes[0],
initial_monitor,
method='monge_ampere',
op=op)
mesh_mover.adapt()
mesh = Mesh(mesh_mover.x)
op.__init__(mesh=mesh, **kwargs)
swp.__init__(op, meshes=[mesh])
# --- Monitor function definitions
def elevation_norm_monitor(mesh, alpha=40.0, norm_type='H1', **kwargs):
"""
Monitor function derived from the elevation `norm_type` norm.
:kwarg alpha: controls the amplitude of the monitor function.
"""
P1DG = FunctionSpace(mesh, "DG", 1)
eta = project(swp.fwd_solutions[0].split()[1], P1DG)
if norm_type == 'hessian_frobenius':
H = recover_hessian(eta, op=op)
return 1.0 + alpha * local_frobenius_norm(H)
# NOTE: We use Monge-Ampere with a monitor function indicating the initial condition
alpha = 10.0 # Parameter controlling prominance of refined region
eps = 1.0e-03 # Parameter controlling width of refined region
def monitor(mesh):
x, y = SpatialCoordinate(mesh)
x0, y0, r = op.source_loc[0]
return conditional(le(abs((x - x0)**2 + (y - y0)**2 - r**2), eps), alpha,
1.0)
mesh_mover = MeshMover(tp.meshes[0], monitor, method='monge_ampere', op=op)
mesh_mover.adapt()
tp.__init__(op, meshes=[Mesh(mesh_mover.x)])
# --- Solve the tracer transport problem
# Note:
# * Pure Lagrangian leads to tangled elements after only a few iterations
# * This motivates applying monitor based methods throughout the simulation
tp.set_initial_condition()
init_vol = assemble(Constant(1.0) * dx(domain=tp.mesh))
init_l1_norm = norm(tp.fwd_solutions_tracer[0], norm_type='L1')
init_l2_norm = norm(tp.fwd_solutions_tracer[0], norm_type='L2')
init_sol = tp.fwd_solutions_tracer[0].copy(deepcopy=True)
tp.solve_forward()
# Compare initial and final tracer concentrations